Distribution Dash Worksheet
Data Detective Field Log
Topic: Normal Distribution Probability Review
Name:
Date:
Instructions: For each problem, identify the mean (\(\mu\)) and standard deviation (\(\sigma\)). Calculate the z-scores and use a standard normal table or calculator to find the probabilities. Show all work in the designated space. Recall that an event is typically considered unusual if its probability is less than 0.05.
01
Light Bulb Lifespan
The lifetimes of a specific brand of LED light bulbs are normally distributed with a mean of 1,500 hours and a standard deviation of 100 hours. A bulb is randomly selected.
(a) Find the probability that the bulb lasts less than 1,450 hours.
(b) Find the probability that the bulb lasts between 1,400 and 1,600 hours.
(c) Find the probability that the bulb lasts more than 1,750 hours. Can this be considered an unusual event? Explain.
02
Cereal Filling Machine
A factory uses a machine to fill cereal boxes. The weights of the cereal are normally distributed with a mean of 16 ounces and a standard deviation of 0.5 ounces. A box is randomly selected.
(a) Find the probability that the box weighs less than 15.2 ounces.
(b) Find the probability that the box weighs between 15.5 and 16.5 ounces.
(c) Find the probability that the box weighs more than 17.2 ounces. Can this be considered an unusual event? Explain.
03
Marathon Completion Times
Completion times for a city marathon are normally distributed with a mean of 240 minutes and a standard deviation of 30 minutes. A runner is randomly selected.
(a) Find the probability that the runner finishes in less than 200 minutes.
(b) Find the probability that the runner finishes between 210 and 250 minutes.
(c) Find the probability that the runner finishes in more than 310 minutes. Can this be considered an unusual event? Explain.
04
Standardized Math Scores
Scores on a statewide math assessment are normally distributed with a mean of 75 and a standard deviation of 8. A student's score is randomly selected.
(a) Find the probability that the student scored less than 60.
(b) Find the probability that the student scored between 70 and 85.
(c) Find the probability that the student scored more than 95. Can this be considered an unusual event? Explain.
05
Douglas Fir Heights
The heights of fully grown Douglas fir trees in a specific forest are normally distributed with a mean of 40 feet and a standard deviation of 5 feet. A tree is randomly selected.
(a) Find the probability that the tree is less than 32 feet tall.
(b) Find the probability that the tree is between 35 and 45 feet tall.
(c) Find the probability that the tree is more than 52 feet tall. Can this be considered an unusual event? Explain.
06
Laptop Battery Life
The battery life of a particular laptop model is normally distributed with a mean of 24 hours and a standard deviation of 3 hours. A laptop is randomly selected.
(a) Find the probability that the battery lasts less than 20 hours.
(b) Find the probability that the battery lasts between 22 and 26 hours.
(c) Find the probability that the battery lasts more than 32 hours. Can this be considered an unusual event? Explain.
07
Morning Commute Time
Commute times for employees in a large city are normally distributed with a mean of 35 minutes and a standard deviation of 7 minutes. A commuter is randomly selected.
(a) Find the probability that the commute is less than 25 minutes.
(b) Find the probability that the commute is between 30 and 45 minutes.
(c) Find the probability that the commute is more than 55 minutes. Can this be considered an unusual event? Explain.
08
Grade A Egg Weights
The weights of Grade A large eggs are normally distributed with a mean of 50 grams and a standard deviation of 4 grams. An egg is randomly selected.
(a) Find the probability that the egg weighs less than 42 grams.
(b) Find the probability that the egg weighs between 45 and 55 grams.
(c) Find the probability that the egg weighs more than 60 grams. Can this be considered an unusual event? Explain.
09
Summer High Temperatures
Daily high temperatures in July for a specific city are normally distributed with a mean of 82°F and a standard deviation of 4°F. A day is randomly selected.
(a) Find the probability that the high temperature is less than 75°F.
(b) Find the probability that the high temperature is between 78°F and 86°F.
(c) Find the probability that the high temperature is more than 92°F. Can this be considered an unusual event? Explain.
10
Smartphone Screen Time
Daily smartphone screen time for teenagers is normally distributed with a mean of 5.5 hours and a standard deviation of 1.2 hours. A teenager is randomly selected.
(a) Find the probability that the screen time is less than 3.5 hours.
(b) Find the probability that the screen time is between 4.5 and 6.5 hours.
(c) Find the probability that the screen time is more than 8.5 hours. Can this be considered an unusual event? Explain.
Distribution Dash Answer Key
Answer Key: Data Detective
Teacher's Edition • Step-by-Step Solutions
Topic: Normal Distribution
Level: Review / Assessment
Instructional Note:
Values below are calculated using standard z-tables. Small variations may occur if using technology (TI-84, Excel). For part (c) in all problems, we use the rule of thumb that an event is "unusual" if its probability is less than 0.05 (5%).
01
Light Bulb Lifespan (\(\mu=1500, \sigma=100\))
(a) Less than 1,450 hours
\(z = \frac{1450 - 1500}{100} = -0.5\)
\(P(X < 1450) = P(Z < -0.5) = 0.3085\)
(b) Between 1,400 and 1,600 hours
\(z_1 = \frac{1400 - 1500}{100} = -1.0\) ; \(z_2 = \frac{1600 - 1500}{100} = 1.0\)
\(P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826\)
(c) More than 1,750 hours
\(z = \frac{1750 - 1500}{100} = 2.5\)
\(P(X > 1750) = 1 - P(Z < 2.5) = 1 - 0.9938 = 0.0062\)
Unusual? Yes. Since \(0.0062 < 0.05\), this is an unusual event.
02
Cereal Filling Machine (\(\mu=16, \sigma=0.5\))
(a) Less than 15.2 ounces
\(z = \frac{15.2 - 16}{0.5} = -1.6\)
\(P(X < 15.2) = P(Z < -1.6) = 0.0548\)
(b) Between 15.5 and 16.5 ounces
\(z_1 = \frac{15.5 - 16}{0.5} = -1.0\) ; \(z_2 = \frac{16.5 - 16}{0.5} = 1.0\)
\(P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826\)
(c) More than 17.2 ounces
\(z = \frac{17.2 - 16}{0.5} = 2.4\)
\(P(X > 17.2) = 1 - 0.9918 = 0.0082\)
Unusual? Yes. Since \(0.0082 < 0.05\), it is unusual.
03
Marathon Completion Times (\(\mu=240, \sigma=30\))
(a) Less than 200 minutes
\(z = \frac{200 - 240}{30} = -1.33\)
\(P(Z < -1.33) = 0.0918\)
(b) Between 210 and 250 minutes
\(z_1 = -1.0, z_2 = 0.33\)
\(P(-1 < Z < 0.33) = 0.6293 - 0.1587 = 0.4706\)
(c) More than 310 minutes
\(z = 2.33 \Rightarrow P(X > 310) = 1 - 0.9901 = 0.0099\)
Unusual? Yes (\(0.0099 < 0.05\)).
04
Standardized Math Scores (\(\mu=75, \sigma=8\))
(a) Less than 60
\(z = \frac{60 - 75}{8} = -1.88\)
\(P(Z < -1.88) = 0.0301\)
(b) Between 70 and 85
\(z_1 = -0.63, z_2 = 1.25\)
\(P(-0.63 < Z < 1.25) = 0.8944 - 0.2643 = 0.6301\)
(c) More than 95
\(z = 2.5 \Rightarrow P(X > 95) = 1 - 0.9938 = 0.0062\)
